This document is about the application of rational choice theory, and is structured as a series of thought experiments to test how individuals make choices in uncertain situations.
After each scenario a short analysis compares the empirical response patterns to predictions generated by economic theory.
As a doctor in a position of authority, you have been informed that a disease will break out in your country next month and result in the deaths of 600 people. (either death or recovery is the outcome in each case). There are two possible vaccination programmes that you can undertake, and undertaking one precludes the other. Programme A will save 400 people with certainty. Programme B will save no one with probability 1/3 and 600 with probability 2/3. Do you prefer programme A or programme B?
As a doctor in a position of authority, you have been informed that a new disease will break out in your country next month. To fight this epidemic, one of two possible vaccination programmes is to be chosen, and undertaking one precludes the other. In programme A, 200 people will die with certainty. In programme B, there is a 2/3 chance that no one will die, and a 1/3 chance that 600 will die. Do you prefer programme A or programme B?
The typical response pattern in experiments of this kind is to choose programme A in round 1, and B in round 2. But the questions are identical in terms of outcomes; the only difference is the manner in which the questions have been phrased. In round 1, 600 people are dead, and most respondents choose to bring some of them back for sure. In round 2, no one is dead yet, and most respondents choose not to consign 200 people to certain death.
You are asked to choose between two gambles. Gamble A gives you £1,000,000 for sure. Gamble B gives you a 10% chance of winning £1,500,000; an 89% chance of winning £1,000,000; and a 1% chance of winning nothing. Do you prefer gamble A or gamble B?
You are asked to choose between two gambles. Gamble A gives you an 11% chance of winning £1,000,000 and an 89% chance of winning nothing. Gamble B gives you a 10% chance of winning £1,500,000 and a 90% chance of winning nothing. Do you prefer gamble A or gamble B?
The Allais Paradox is named after Maurice Allais, and is the most famous violation of the von Neumann-Morgenstern expected utility theory. The typical response pattern in studies of this type is to take the sure thing in the first round, and the second gamble in the second (A,B). This contradicts the substituion axiom of VNM expected utility theory (sometimes known as the independence axiom, as it refers to the independence of irrelevant alternatives). When working with expected utility, if two mutually exclusive events A and B generate utility u(A) with probability p and u(B) with probability q, we can write:
Expected utility = p·u(A) + q·u(B)
It is now straighforward to see why choosing (A,B) in this example is inconsistent with expected utility maximisation. If we write the indirect utility function as v(·) and if A is preferred to B in round 1, then:
Expected utility of gamble A > Expected utility of gamble B (round 1)
v(1,000,000) > 0.10·v(1,500,000) + 0.89·v(1,000,000) + 0.01·v(0)
Which after collecting the v(1,000,000) terms is:
0.11·v(1,000,000) > 0.10·v(1,500,000) + 0.01·v(0)
Now add 0.89·v(0) to both sides:
0.11·v(1,000,000) + 0.89·v(0) > 0.10·v(1,500,000) + 0.90·v(0)
But this corresponds exactly to the choice of gambles available in round 2, so:
Expected utility of gamble A > Expected utility of gamble B (round 2)
And thus if the substitution axiom holds, gamble A should also be preferred to gamble B in round 2. Equivalently, mixing the each of the two gambles in round 1 with a third (and so generating the gambles in round 2) should not change an individual's preference ordering. The converse also holds; if B is preferred in round 1, it should also be preferred in round 2.
An urn contains 300 coloured marbles. 100 of the marbles are red, the remaining 200 are some mixture of blue and green. A marble will be selected at random from the urn. You will receive £1,000 if the marble selected is of a specified colour. Would you rather that colour be red or blue?
An urn contains 300 coloured marbles. 100 of the marbles are red, the remaining 200 are some mixture of blue and green. A marble will be selected at random from the urn. You will receive £1,000 if the marble selected is not of a specified colour. Would you rather that colour be red or blue?
The Ellsberg Paradox is named after Daniel Ellsberg, who found that the typical response pattern in studies of this type is to choose red in both cases. This response pattern violates the Savage or Anscombe-Aumann axioms. When these are satisfied, choices are made as if subjective probabilities are assessed over the colour of the marble. If red is preferred in round 1, the decision maker must assess a higher probability for red than blue. If this is the case, he should then assess a higher probability for not blue than not red, and prefer to answer blue in round 2. The converse applies if blue is preferred in the first round.